Sullivan completions

TL;DR

This paper surveys the properties of Sullivan completions, a construction in algebraic topology that associates a minimal model to a space and explores its geometric realization.

Contribution

It provides a comprehensive overview of Sullivan completions, including explicit examples and key properties, clarifying their role in rational homotopy theory.

Findings

Sullivan completions relate spaces to their minimal models.

Explicit examples illustrate the construction and properties.

The survey clarifies the significance of Sullivan completions in topology.

Abstract

The Sullivan construction associates to each path connected space or connected simplicial set, $X$, a special cdga, its minimal model $(\land V,d)$, and to each such cdga $\land W$ its geometric realisation $\langle \land W\rangle$. The composite of these constructions is the Sullivan completion, $X_{\mathbb Q}$, of $X$. In this paper we give a survey of the main properties of Sullivan completions, and include explicit examples.